I am very glad to hear that your boyfriend has decided to seek therapy.
An insecure person will often appear to favor the New Shiny person who pays a lot of attention to them, because it strokes their ego, while the existing partner gets ignored, or close to it. Such people may pursue new relationships on a level that looks obsessive to their existing partner or partners. I give your metamour a lot of credit for not just blaming you for his bad behavior.
Kudos for you too for offering to babysit so they can go out. That was extremely nice of you.
I hope things work out for the three of you.
Female, married just about forever to Scout. My boyfriends are E. and C. Both are married, and C also has another lover, J.